Mapped texture not showing in viewport
For cycles, once you have your object unwrapped and textured like you have in your first screenshot, you then navigate to the material tab, and click "Use Nodes"

Now under "Surface" you should have a drop down which says "Diffuse BSDF", followed by 3 options; "Color", "Rough", and "Norm". Click the little dot to the right of the "Color" widget, and choose "Image Texture":



In the image texture panel, click the drop down to the left of the "Open" button, and select the image that you had open in the UV editor:



Now if you go into the textured view (Alt+Z), you should see your texture applied to your object.